[math-fun] Coordination sequences for polyhedra
Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed. Tom
Dear Tom Karzes , For those polyhedra that you mention, some of the coordination sequences are too short (like 1 3 3 1 for the cube) or too similar to other sequences; and some of the others, the easier ones, I have done by hand, but I haven't done them all - and I would certainly like to get them. Maybe the simplest way to do this would be for you to do them all (more-or-less! - if there are infinite series like the prisms then just do the first handful), and send them to me, and I'll make the decision about which ones are too short, etc., and then I will add the ones I don't have to the OEIS (giving you credit of course) I'm glad you suggested this - it is something that has needed doing for a long time. And don't feel that you need to restrict this to the classical solids that you mentioned. Non-convex solids are also worth doing, even solids with holes. That is opening up a large can of worms, of course - so if other folks want to help then by all means join in. I remember that many years ago Tom Duff (on this list) had a catalog at Bell Labs of a great many solids and their properties. I still have the TM somewhere. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Mon, Jan 6, 2020 at 3:22 PM Tom Karzes <karzes@sonic.net> wrote:
Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
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I'll look at this later tonight (it's early afternoon where I am). I have all the data, but I'll need to massage it a little to get it into the form I need, and I'll need to display the results in a satisfying manner. But once I have it set up, handling all of the cases will be no harder than handling one of them. I hadn't really thought about the infinite (planar) cases, but with a bit more work I can probably do something for those as well. At the moment, I handle them as toroidal tilings, with a horizontal and vertical replication factor. Tom Neil Sloane writes:
Dear Tom Karzes , For those polyhedra that you mention, some of the coordination sequences are too short (like 1 3 3 1 for the cube) or too similar to other sequences; and some of the others, the easier ones, I have done by hand, but I haven't done them all - and I would certainly like to get them.
Maybe the simplest way to do this would be for you to do them all (more-or-less! - if there are infinite series like the prisms then just do the first handful), and send them to me, and I'll make the decision about which ones are too short, etc., and then I will add the ones I don't have to the OEIS (giving you credit of course)
I'm glad you suggested this - it is something that has needed doing for a long time.
And don't feel that you need to restrict this to the classical solids that you mentioned. Non-convex solids are also worth doing, even solids with holes.
That is opening up a large can of worms, of course - so if other folks want to help then by all means join in.
I remember that many years ago Tom Duff (on this list) had a catalog at Bell Labs of a great many solids and their properties. I still have the TM somewhere.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Mon, Jan 6, 2020 at 3:22 PM Tom Karzes <karzes@sonic.net> wrote:
Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
Well I didn't get to it last night, but I did get to it today. The following page contains coordination sequences for all of the Platonic, Archimedean, and Catalan solids. It also contains links in the left column to interactive images (if you have a mouse, you can click and drag to manually rotate them, or you can let them spin by themselves): https://www.karzes.com/polyhedra/cs_info.html Tom Tom Karzes writes:
I'll look at this later tonight (it's early afternoon where I am). I have all the data, but I'll need to massage it a little to get it into the form I need, and I'll need to display the results in a satisfying manner. But once I have it set up, handling all of the cases will be no harder than handling one of them.
I hadn't really thought about the infinite (planar) cases, but with a bit more work I can probably do something for those as well. At the moment, I handle them as toroidal tilings, with a horizontal and vertical replication factor.
Tom
Neil Sloane writes:
Dear Tom Karzes , For those polyhedra that you mention, some of the coordination sequences are too short (like 1 3 3 1 for the cube) or too similar to other sequences; and some of the others, the easier ones, I have done by hand, but I haven't done them all - and I would certainly like to get them.
Maybe the simplest way to do this would be for you to do them all (more-or-less! - if there are infinite series like the prisms then just do the first handful), and send them to me, and I'll make the decision about which ones are too short, etc., and then I will add the ones I don't have to the OEIS (giving you credit of course)
I'm glad you suggested this - it is something that has needed doing for a long time.
And don't feel that you need to restrict this to the classical solids that you mentioned. Non-convex solids are also worth doing, even solids with holes.
That is opening up a large can of worms, of course - so if other folks want to help then by all means join in.
I remember that many years ago Tom Duff (on this list) had a catalog at Bell Labs of a great many solids and their properties. I still have the TM somewhere.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Mon, Jan 6, 2020 at 3:22 PM Tom Karzes <karzes@sonic.net> wrote:
Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
Functional and looks nice. All in Javascript? WFL On 1/8/20, Tom Karzes <karzes@sonic.net> wrote:
Well I didn't get to it last night, but I did get to it today.
The following page contains coordination sequences for all of the Platonic, Archimedean, and Catalan solids. It also contains links in the left column to interactive images (if you have a mouse, you can click and drag to manually rotate them, or you can let them spin by themselves):
https://www.karzes.com/polyhedra/cs_info.html
Tom
Tom Karzes writes:
I'll look at this later tonight (it's early afternoon where I am). I have all the data, but I'll need to massage it a little to get it into the form I need, and I'll need to display the results in a satisfying manner. But once I have it set up, handling all of the cases will be no harder than handling one of them.
I hadn't really thought about the infinite (planar) cases, but with a bit more work I can probably do something for those as well. At the moment, I handle them as toroidal tilings, with a horizontal and vertical replication factor.
Tom
Neil Sloane writes:
Dear Tom Karzes , For those polyhedra that you mention, some of the coordination sequences are too short (like 1 3 3 1 for the cube) or too similar to other sequences; and some of the others, the easier ones, I have done by hand, but I haven't done them all - and I would certainly like to get them.
Maybe the simplest way to do this would be for you to do them all (more-or-less! - if there are infinite series like the prisms then just do the first handful), and send them to me, and I'll make the decision about which ones are too short, etc., and then I will add the ones I don't have to the OEIS (giving you credit of course)
I'm glad you suggested this - it is something that has needed doing for a long time.
And don't feel that you need to restrict this to the classical solids that you mentioned. Non-convex solids are also worth doing, even solids with holes.
That is opening up a large can of worms, of course - so if other folks want to help then by all means join in.
I remember that many years ago Tom Duff (on this list) had a catalog at Bell Labs of a great many solids and their properties. I still have the TM somewhere.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Mon, Jan 6, 2020 at 3:22 PM Tom Karzes <karzes@sonic.net> wrote:
Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
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Thanks Fred. The web pages are generated by Python scripts. The coordination sequence page is just a static HTML page with links. The links point to an interactive graphical page which does make heavy use of JavaScript. So the static portions are generated by Python, and the dynamic portions use JavaScript. Neil, was this the information you were looking for (the coordination sequences)? Tom Fred Lunnon writes:
Functional and looks nice. All in Javascript? WFL
On 1/8/20, Tom Karzes <karzes@sonic.net> wrote:
Well I didn't get to it last night, but I did get to it today.
The following page contains coordination sequences for all of the Platonic, Archimedean, and Catalan solids. It also contains links in the left column to interactive images (if you have a mouse, you can click and drag to manually rotate them, or you can let them spin by themselves):
https://www.karzes.com/polyhedra/cs_info.html
Tom
Tom Karzes writes:
I'll look at this later tonight (it's early afternoon where I am). I have all the data, but I'll need to massage it a little to get it into the form I need, and I'll need to display the results in a satisfying manner. But once I have it set up, handling all of the cases will be no harder than handling one of them.
I hadn't really thought about the infinite (planar) cases, but with a bit more work I can probably do something for those as well. At the moment, I handle them as toroidal tilings, with a horizontal and vertical replication factor.
Tom
Neil Sloane writes:
Dear Tom Karzes , For those polyhedra that you mention, some of the coordination sequences are too short (like 1 3 3 1 for the cube) or too similar to other sequences; and some of the others, the easier ones, I have done by hand, but I haven't done them all - and I would certainly like to get them.
Maybe the simplest way to do this would be for you to do them all (more-or-less! - if there are infinite series like the prisms then just do the first handful), and send them to me, and I'll make the decision about which ones are too short, etc., and then I will add the ones I don't have to the OEIS (giving you credit of course)
I'm glad you suggested this - it is something that has needed doing for a long time.
And don't feel that you need to restrict this to the classical solids that you mentioned. Non-convex solids are also worth doing, even solids with holes.
That is opening up a large can of worms, of course - so if other folks want to help then by all means join in.
I remember that many years ago Tom Duff (on this list) had a catalog at Bell Labs of a great many solids and their properties. I still have the TM somewhere.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Mon, Jan 6, 2020 at 3:22 PM Tom Karzes <karzes@sonic.net> wrote:
Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
But it looks like the JS that does the three dimensional projection is all yours, is that right? The td/ scripts? Is that written up anywhere? On Tue, Jan 7, 2020 at 8:04 PM Tom Karzes <karzes@sonic.net> wrote:
Thanks Fred. The web pages are generated by Python scripts. The coordination sequence page is just a static HTML page with links. The links point to an interactive graphical page which does make heavy use of JavaScript. So the static portions are generated by Python, and the dynamic portions use JavaScript.
Neil, was this the information you were looking for (the coordination sequences)?
Tom
Fred Lunnon writes:
Functional and looks nice. All in Javascript? WFL
On 1/8/20, Tom Karzes <karzes@sonic.net> wrote:
Well I didn't get to it last night, but I did get to it today.
The following page contains coordination sequences for all of the Platonic, Archimedean, and Catalan solids. It also contains links in the left column to interactive images (if you have a mouse, you can click and drag to manually rotate them, or you can let them spin by themselves):
https://www.karzes.com/polyhedra/cs_info.html
Tom
Tom Karzes writes:
I'll look at this later tonight (it's early afternoon where I am). I have all the data, but I'll need to massage it a little to get it into the form I need, and I'll need to display the results in a satisfying manner. But once I have it set up, handling all of the cases will be no harder than handling one of them.
I hadn't really thought about the infinite (planar) cases, but with a bit more work I can probably do something for those as well. At the moment, I handle them as toroidal tilings, with a horizontal and vertical replication factor.
Tom
Neil Sloane writes:
Dear Tom Karzes , For those polyhedra that you mention, some of the coordination sequences are too short (like 1 3 3 1 for the cube) or too similar to other sequences; and some of the others, the easier ones, I have done by hand, but I haven't done them all - and I would certainly like to get them.
Maybe the simplest way to do this would be for you to do them all (more-or-less! - if there are infinite series like the prisms then just do the first handful), and send them to me, and I'll make the decision about which ones are too short, etc., and then I will add the ones I don't have to the OEIS (giving you credit of course)
I'm glad you suggested this - it is something that has needed doing for a long time.
And don't feel that you need to restrict this to the classical solids that you mentioned. Non-convex solids are also worth doing, even solids with holes.
That is opening up a large can of worms, of course - so if other folks want to help then by all means join in.
I remember that many years ago Tom Duff (on this list) had a catalog at Bell Labs of a great many solids and their properties. I still have the TM somewhere.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Mon, Jan 6, 2020 at 3:22 PM Tom Karzes <karzes@sonic.net> wrote:
Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
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I wrote all of the scripts and libraries myself. I haven't exported them for general use. I pretty much only use them for the polyhedron stuff. Tom Tomas Rokicki writes:
But it looks like the JS that does the three dimensional projection is all yours, is that right? The td/ scripts? Is that written up anywhere?
On Tue, Jan 7, 2020 at 8:04 PM Tom Karzes <karzes@sonic.net> wrote:
Thanks Fred. The web pages are generated by Python scripts. The coordination sequence page is just a static HTML page with links. The links point to an interactive graphical page which does make heavy use of JavaScript. So the static portions are generated by Python, and the dynamic portions use JavaScript.
Neil, was this the information you were looking for (the coordination sequences)?
Tom
Fred Lunnon writes:
Functional and looks nice. All in Javascript? WFL
On 1/8/20, Tom Karzes <karzes@sonic.net> wrote:
Well I didn't get to it last night, but I did get to it today.
The following page contains coordination sequences for all of the Platonic, Archimedean, and Catalan solids. It also contains links in the left column to interactive images (if you have a mouse, you can click and drag to manually rotate them, or you can let them spin by themselves):
https://www.karzes.com/polyhedra/cs_info.html
Tom
Tom Karzes writes:
I'll look at this later tonight (it's early afternoon where I am). I have all the data, but I'll need to massage it a little to get it into the form I need, and I'll need to display the results in a satisfying manner. But once I have it set up, handling all of the cases will be no harder than handling one of them.
I hadn't really thought about the infinite (planar) cases, but with a bit more work I can probably do something for those as well. At the moment, I handle them as toroidal tilings, with a horizontal and vertical replication factor.
Tom
Neil Sloane writes:
Dear Tom Karzes , For those polyhedra that you mention, some of the coordination sequences are too short (like 1 3 3 1 for the cube) or too similar to other sequences; and some of the others, the easier ones, I have done by hand, but I haven't done them all - and I would certainly like to get them.
Maybe the simplest way to do this would be for you to do them all (more-or-less! - if there are infinite series like the prisms then just do the first handful), and send them to me, and I'll make the decision about which ones are too short, etc., and then I will add the ones I don't have to the OEIS (giving you credit of course)
I'm glad you suggested this - it is something that has needed doing for a long time.
And don't feel that you need to restrict this to the classical solids that you mentioned. Non-convex solids are also worth doing, even solids with holes.
That is opening up a large can of worms, of course - so if other folks want to help then by all means join in.
I remember that many years ago Tom Duff (on this list) had a catalog at Bell Labs of a great many solids and their properties. I still have the TM somewhere.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Mon, Jan 6, 2020 at 3:22 PM Tom Karzes <karzes@sonic.net> wrote:
> Neil, do you have coordination sequences for all of the > Platonic/Archimedean/Catalan solids? Those seem like the > most fundamental ones for polyhedra. I could probably > generate them without too much trouble if needed. > > Tom >
To Tom Karzes: Thanks for computing those sequences! I'll process them tomorrow Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Tue, Jan 7, 2020 at 11:22 PM Tom Karzes <karzes@sonic.net> wrote:
I wrote all of the scripts and libraries myself. I haven't exported them for general use. I pretty much only use them for the polyhedron stuff.
Tom
Tomas Rokicki writes:
But it looks like the JS that does the three dimensional projection is all yours, is that right? The td/ scripts? Is that written up anywhere?
On Tue, Jan 7, 2020 at 8:04 PM Tom Karzes <karzes@sonic.net> wrote:
Thanks Fred. The web pages are generated by Python scripts. The coordination sequence page is just a static HTML page with links. The links point to an interactive graphical page which does make heavy use of JavaScript. So the static portions are generated by Python, and the dynamic portions use JavaScript.
Neil, was this the information you were looking for (the coordination sequences)?
Tom
Fred Lunnon writes:
Functional and looks nice. All in Javascript? WFL
On 1/8/20, Tom Karzes <karzes@sonic.net> wrote:
Well I didn't get to it last night, but I did get to it today.
The following page contains coordination sequences for all of the Platonic, Archimedean, and Catalan solids. It also contains links in the left column to interactive images (if you have a mouse, you can click and drag to manually rotate them, or you can let them spin by themselves):
https://www.karzes.com/polyhedra/cs_info.html
Tom
Tom Karzes writes:
I'll look at this later tonight (it's early afternoon where I am). I have all the data, but I'll need to massage it a little to get it into the form I need, and I'll need to display the results in a satisfying manner. But once I have it set up, handling all of the cases will be no harder than handling one of them.
I hadn't really thought about the infinite (planar) cases, but with a bit more work I can probably do something for those as well. At the moment, I handle them as toroidal tilings, with a horizontal and vertical replication factor.
Tom
Neil Sloane writes: > Dear Tom Karzes , > For those polyhedra that you mention, > some of the coordination sequences are too short (like 1 3 3 1 for the cube) > or too similar to other sequences; > and some of the others, the easier ones, I have done by hand, > but I haven't done them all - and I would certainly like to get them. > > Maybe the simplest way to do this would be for you to do them all > (more-or-less! > - if there are infinite series like the prisms then just do the first > handful), > and send them to me, and I'll make the decision about which > ones are too short, etc., and then I will add the ones I don't have to the > OEIS (giving you credit of course) > > I'm glad you suggested this - it is something that has needed doing for a > long time. > > And don't feel that you need to restrict this to the classical solids that > you mentioned. > Non-convex solids are also worth doing, even solids with holes. > > That is opening up a large can of worms, of course - so if > other folks want to help then by all means join in. > > I remember that many years ago Tom Duff (on this list) had > a catalog at Bell Labs of a great many solids and their properties. I still > have the TM somewhere. > > Best regards > Neil > > Neil J. A. Sloane, President, OEIS Foundation. > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. > Phone: 732 828 6098; home page: http://NeilSloane.com > Email: njasloane@gmail.com > > > > On Mon, Jan 6, 2020 at 3:22 PM Tom Karzes < karzes@sonic.net> wrote: > > > Neil, do you have coordination sequences for all of the > > Platonic/Archimedean/Catalan solids? Those seem like the > > most fundamental ones for polyhedra. I could probably > > generate them without too much trouble if needed. > > > > Tom > >
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On Mon, Jan 6, 2020 at 12:54 PM Neil Sloane <njasloane@gmail.com> wrote:
I remember that many years ago Tom Duff (on this list) had a catalog at Bell Labs of a great many solids and their properties. I still have the TM somewhere. It wasn't me, though I followed the work with interest -- Andrew Hume, I think.
Has anybody considered coordination trees (as it were) for aperiodic tilings (sic --- prefer quasi-crystallographic!), such as planar Penrose rhombs and its generalisations to solid honeycombs? I gather that these might be of interest to crystallographers, without understanding details of the applications. WFL On 1/6/20, Tom Karzes <karzes@sonic.net> wrote:
Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
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That's an interesting idea Fred. If I understand you correctly, you're proposing creating branch points where there's more than one choice for an adjacent tile (which doesn't occur for periodic tilings). It sounds difficult. I suppose for each new tile you could create branches for each set of possible adjacent tiles. One difficulty is that, proceeding in this manner, it is possible to create dead-end configurations that don't tile the entire plane. Perhaps using the inflation rules might be more deterministic. You'd just need to look at a large enough area to ensure that you've found every possible pattern radiating from a given starting configuration. Tom Fred Lunnon writes:
Has anybody considered coordination trees (as it were) for aperiodic tilings (sic --- prefer quasi-crystallographic!), such as planar Penrose rhombs and its generalisations to solid honeycombs?
I gather that these might be of interest to crystallographers, without understanding details of the applications.
WFL
On 1/6/20, Tom Karzes <karzes@sonic.net> wrote:
Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
For substitution tilings, there are many ways to derive a tree structure from the rules themselves, and I think there are canonical encodings in the topological theory, see for example Lorenzo Sadun “Topology of Tiling Spaces”. The half-hex tiling, for example, has the topology of a Quadtree. You may have already seen that quadtrees are sometimes used to describe snowflake growth, for image compression, or possibly for image scrambling. I also searched OEIS, and found: https://oeis.org/search?q=Penrose+coordination+&language=english&go=Search There are three dissimilar entries for 5-fold coordination sequences, but only one 5-fold fixed point of the Penrose tiling. It alternates with period 2, between sun and star. This explains two of the three entries, what about the third?? https://oeis.org/A302176/a302176_1.png The vertex at n=2 appears to be illegal by Penrose’s matching rules, so I don’t know why the name says “Penrose tiling”. Maybe a comment should he added saying: “This is not a Penrose Tiling”? And if the rules don’t hold, how is the pattern expanded? The reference is paywalled, so that is not helping... —Brad
On Jan 6, 2020, at 3:02 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Has anybody considered coordination trees (as it were) for aperiodic tilings (sic --- prefer quasi-crystallographic!), such as planar Penrose rhombs and its generalisations to solid honeycombs?
I gather that these might be of interest to crystallographers, without understanding details of the applications.
WFL
On 1/6/20, Tom Karzes <karzes@sonic.net> wrote: Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
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I just sent Brad Klee a copy of the article that A302176 is based on, and I'll be happy to send a copy to anyone else who is interested. This has to do with a coordination sequence for a vertex in a certain Penrose tiling Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Mon, Jan 6, 2020 at 5:53 PM Brad Klee <bradklee@gmail.com> wrote:
For substitution tilings, there are many ways to derive a tree structure from the rules themselves, and I think there are canonical encodings in the topological theory, see for example Lorenzo Sadun “Topology of Tiling Spaces”.
The half-hex tiling, for example, has the topology of a Quadtree. You may have already seen that quadtrees are sometimes used to describe snowflake growth, for image compression, or possibly for image scrambling.
I also searched OEIS, and found: https://oeis.org/search?q=Penrose+coordination+&language=english&go=Search
There are three dissimilar entries for 5-fold coordination sequences, but only one 5-fold fixed point of the Penrose tiling. It alternates with period 2, between sun and star. This explains two of the three entries, what about the third??
https://oeis.org/A302176/a302176_1.png
The vertex at n=2 appears to be illegal by Penrose’s matching rules, so I don’t know why the name says “Penrose tiling”. Maybe a comment should he added saying: “This is not a Penrose Tiling”? And if the rules don’t hold, how is the pattern expanded? The reference is paywalled, so that is not helping...
—Brad
On Jan 6, 2020, at 3:02 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Has anybody considered coordination trees (as it were) for aperiodic tilings (sic --- prefer quasi-crystallographic!), such as planar Penrose rhombs and its generalisations to solid honeycombs?
I gather that these might be of interest to crystallographers, without understanding details of the applications.
WFL
On 1/6/20, Tom Karzes <karzes@sonic.net> wrote: Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
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To answer Fred's question: we have something like ten thousand coord sequences for 3-D crystals (mostly periodic structures, naturally) there are maybe a dozen that arise from planar aperiodic tilings (Penrose, as Brad discovered, Ammann-Beenker, and a few others) But IIRC we don't have any for 3-D Penrose-type structures. Wish we did! We do have a lot from this web site: Reticular Chemistry Structure Resource (RCSR), http://rcsr.net and also from the ToposPro web site: http://www.topospro.com On Mon, Jan 6, 2020 at 4:02 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Has anybody considered coordination trees (as it were) for aperiodic tilings (sic --- prefer quasi-crystallographic!), such as planar Penrose rhombs and its generalisations to solid honeycombs?
I gather that these might be of interest to crystallographers, without understanding details of the applications.
WFL
On 1/6/20, Tom Karzes <karzes@sonic.net> wrote:
Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
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participants (6)
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Brad Klee -
Fred Lunnon -
Neil Sloane -
Tom Duff -
Tom Karzes -
Tomas Rokicki